on the uncertainty of real-time predictions of epidemic

HAL Id: hal-02894992https://hal.archives-ouvertes.fr/hal-02894992

Submitted on 9 Jul 2020

HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.

On the uncertainty of real-time predictions of epidemicgrowths: A COVID-19 case study for China and Italy

Tommaso Alberti, Davide Faranda

To cite this version:Tommaso Alberti, Davide Faranda. On the uncertainty of real-time predictions of epidemic growths:A COVID-19 case study for China and Italy. Communications in Nonlinear Science and NumericalSimulation, Elsevier, 2020, 90, pp.105372. �10.1016/j.cnsns.2020.105372�. �hal-02894992�

https://hal.archives-ouvertes.fr/hal-02894992

https://hal.archives-ouvertes.fr

On the uncertainty of real-time predictions of epidemic

growths: a COVID-19 case study for China and Italy

Tommaso Albertia, Davide Farandab,c,d

aINAF - Istituto di Astrofisica e Planetologia Spaziali, via del Fosso del Cavaliere 100,00133 Roma, Italy

bLaboratoire des Sciences du Climat et de l’Environnement, 5 CEA Saclay l’Orme desMerisiers, UMR 8212 CEA-CNRS-UVSQ, 6 Universite Paris-Saclay & IPSL, 91191

Gif-sur-Yvette, FrancecLondon Mathematical Laboratory, 8 Margravine Gardens, London, W6 8RH, UKdLMD/IPSL, Ecole Normale Superieure, 9 PSL research University, Paris, France

Abstract

While COVID-19 is rapidly propagating around the globe, the need for pro-viding real-time forecasts of the epidemics pushes fits of dynamical and sta-tistical models to available data beyond their capabilities. Here we focus onstatistical predictions of COVID-19 infections performed by fitting asymp-totic distributions to actual data. By taking as a case-study the epidemicevolution of total COVID-19 infections in Chinese provinces and Italian re-gions, we find that predictions are characterized by large uncertainties atthe early stages of the epidemic growth. Those uncertainties significantlyreduce after the epidemics peak is reached. Differences in the uncertaintyof the forecasts at a regional level can be used to highlight the delay in thespread of the virus. Our results warn that long term extrapolation of epi-demics counts must be handled with extreme care as they crucially dependnot only on the quality of data, but also on the stage of the epidemics, dueto the intrinsically non-linear nature of the underlying dynamics. These re-sults suggest that real-time epidemiological projections should include wideuncertainty ranges and urge for the needs of compiling high-quality datasetsof infections counts, including asymptomatic patients.

Keywords: COVID-19, Logistic model, Epidemic model, National vs.Regional diffusion

Preprint submitted to Communications in Nonlinear Science and Numerical SimulationJuly 9, 2020

1. Introduction1

The COVID-19, a disease caused by the SARS-CoV-2 virus, was firstly2

reported in the Hubei province on 31 December 2019 when the WHO China3

Country Office was informed of cases of pneumonia unknown etiology de-4

tected in Wuhan City [1, 2, 3]. On 7 January 2020 the Chinese authorities5

identified this virus as a zoonotic virus belonging to the family of coronavirus6

[4, 5, 6]. Its diffusion rapidly spread over all Chinese provinces and near-7

est countries (Thailand, Japan, Korea) [7]. On 23 January, although still8

unknown the initial source of the epidemic, the evidence that 2019-nCoV9

spreads from human-to-human and also across generations of cases quickly10

increases [8, 9]. On 30 January, the World Health Organization (WHO)11

declared the outbreak to be a public health emergency of international con-12

cern [10], believing that it is still possible to interrupt the virus spread by13

putting in place strong measures for early detecting, isolating, and treating14

cases, for tracing back all contacts, and for promoting social distancing mea-15

sures [10, 11, 12]. The main driver of transmission is still an open question16

[13, 14], and preliminary estimates of the median incubation period are 5-617

days (ranging between 2 and 14 days) [15]. On 21 February a cluster of cases18

was detected in Italy (Lombardia), then on 23 February 11 municipalities19

in northern Italy were identified as the two main Italian clusters and placed20

under quarantine [16], on 9 March the quarantine has been expanded to all21

of Italy [17], on 11 March all commercial activity except for supermarkets22

and pharmacies were prohibited [18], and on 22 March all non-essential busi-23

nesses and industries were closed [19] and additional restrictions to movement24

of people were introduced [20, 21].25

Meanwhile, the quarantined Chinese regions observed a fast decrease in26

the number of cases in Hubei and a moderate decrease in other affected re-27

gions, at the same time the virus internationally spread, and on 11 March the28

WHO declared COVID-19 a pandemic [22, 23]. To date, there are more than29

1 million confirmed cases over the globe, more than 60000 deaths, and the30

most affected areas are the European region and the United States. While31

three months were needed to reach the first 100000 confirmed cases, only 2332

days were sufficient to multiply by eight the counts, a typical signature of the33

exponential spreading of viruses. The reason for such high infectivity are cur-34

rently being explored in clinical studies and numerical simulations [24]. Due35

to the fast spread of the virus and the severity of symptoms, restrictive con-36

finement measures have been imposed in many countries. They were based37

2

on asymptotic extrapolation of infection counts obtained on the basis of com-38

partmental epidemic models as the Susceptible-Exposed-Infected-Recovered39

(SEIR) model and their variants [25] or on agent-based models [26]. Unfortu-40

nately, predictions made using these models are extremely sensitive to the un-41

derlying parameters and the quality of their extrapolation is deeply affected42

from both the lack of high-quality datasets as well as from the intrinsic sen-43

sitivity of the dynamics to initial conditions in the growing phase [27]. More-44

over, in order to provide reliable estimates of asymptotic infection counts, a45

knowledge of asymptomatic populations is needed. These data are currently46

almost unavailable and affected by great uncertainties.47

Another possibility is to extrapolate the number of infections by means48

of fitting asymptotic distributions to actual data. Using these phenomeno-49

logical statistical approach, we compare the behavior of epidemic evolution50

across China and Italy. The assumption beyond those fits is that typical51

curves of total infections in SEIR models display a sigmoid shape [28]. Sig-52

moid functions such as the logistic or Gompertz can therefore be used to53

fit actual data. When data are collected with the same protocols, e.g., in54

China and Italy, where tests are performed only to symptomatic patients,55

the statistical fitting can therefore provide an extrapolation of how many56

symptomatic cases should be recorded, although it will not inform about57

the real percentage of infected population [29]. We found that predictions58

are characterized by large uncertainties at the early stages of the epidemic59

growth, significantly reducing when a mature stage or a peak of infections60

are reached. This is observed both in China and in Italy, although some61

differences are observed across the Italian territory, possibly related with the62

time delayed diffusion of epidemic into the different Italian regions. Finally,63

we also estimate infection increments for each Italian region, with being the64

uncertainty significantly reduced for Northern and Central regions, while a65

larger one is found for Southern regions. These results can be helpful for66

each epidemic diffusion, thus highlighting that the confinement measures are67

fundamental and more effective in the early stages of the epidemic evolution68

(the first 7 days), thus producing a different spread across provinces/regions69

as these measures are considered. The main novelty introduced in this70

work is to investigate how uncertainty changes during the different71

stages of the epidemics. This is a crucial aspect that needs to be72

carefully considered when long-term extrapolations of the infection73

counts are carried out since they significantly depend not only on74

the quality of data, but also on the stage of the epidemics, due75

3

to the intrinsically non-linear nature of the underlying dynamics.76

This has also profound consequences on modeling epidemic growth77

by means of dynamical models as those based on compartments78

or agent dynamics which need to be initialized with quality data,79

faithfully representing the infected populations including asymp-80

tomatic patients [27]. Our approach, based on a sort of Bayesian81

framework to reduce uncertainty as more data and/or information82

become available, is particularly helpful for unknown viruses and83

outbreaks, and allows to suggest few practical guidelines to con-84

trol the local diffusion of epidemics and to restrict the analysis85

on specific regions, aiming at preserving the public health and at86

enforcing/relaxing confinement measures.87

2. Data88

Data for the Chinese provinces are obtained from the data repository for89

the 2019 Novel Coronavirus Visual Dashboard operated by the Johns Hop-90

kins University Center for Systems Science and Engineering (JHU CSSE),91

freely available at https://github.com/CSSEGISandData/COVID-19. Fig. 192

reports the total number of confirmed infections (left panel), thus including93

actual positive people to COVID-19, recovered and deaths for China and94

three Chinese provinces of Bejing, Hubei, and Yunnan, and the daily infec-95

tions (right panel), during the period between 22 January and 30 March.96

97

Data for the Italian regions are instead derived from the repository freely98

available at https://github.com/pcm-dpc/COVID-19 where data are col-99

lected from the Italian Protezione Civile from 24 February 2020. Data used100

here were last downloaded on 02 April, thus covering the period 24 February101

- 02 April, as shown in Fig. 2. It is evident that although the increments of102

infections started about 1 month after the Chinese epidemic Italy has fast103

reached and exceeded the Chinese peak values of ∼80000 infections. More-104

over, it is also apparent that epidemic diffusion in China reached its peak105

within ∼20 days from the first restriction operated to the Hubei region on 23106

January. Conversely, the Italian restrictions seem to become more efficient107

only when the Italian government adopted a lock-down confinement on 9108

March [17].109

4

30 Jan 19 Feb 10 Mar 30 Mar 10

0

101

102

103

104

105

Co

nfi

rmed

cas

es

China

Hubei

Bejing

Yunnan

30 Jan 19 Feb 10 Mar 30 Mar 10

0

101

102

103

104

105

Dai

ly i

nfe

ctio

ns

Figure 1: The total number of confirmed infections (left panel) and the daily infections(right panel) for China and three Chinese provinces of Bejing, Hubei, and Yunnan. Filledcircles refer to the first 30 days of the epidemic diffusion. The vertical dashed lines markthe times when Chinese government applied lock-down restrictions on 23 January and 28January, respectively.

3. Methods110

A data-driven way to extrapolate future phases, in terms of both key111

parameters and epidemic impact, of an epidemic growth [30, 31, 32] is112

to use phenomenological statistical models [33]. Indeed, since the113

total number of infections C(t) is a sigmoid function different kinds114

of models can be used to fit its time evolution [34]. Within the115

large variety of possible sigmoid functions the generalized logistic116

distribution and the generalized Gompertz one have proven to be117

successful in fitting epidemic growths [35, 36]. Their suitability118

is mostly related to the reduced number of free parameters (only119

three) with respect to other choices depending on a larger set of120

model parameters (>3) which allows to reduce the overfitting ef-121

fect due to a statistical model containing more parameters than122

can be justified by the data [34]. However, our main aim of inves-123

tigating how uncertainty evolves with the epidemic growth stage124

is independent on the choice of the fitting distribution provided125

that they are dependent on the same number of free parameters.126

5

20 Feb 29 Feb 10 Mar 20 Mar 30 Mar 09 Apr10

0

101

102

103

104

105

Co

nfi

rmed

cas

es

Italy

Lombardia

Marche

Puglia

20 Feb 29 Feb 10 Mar 20 Mar 30 Mar 09 Apr10

0

101

102

103

104

Dai

ly i

nfe

ctio

ns

Figure 2: The total number of confirmed infections (left panel) and the daily infections(right panel) for Italy and three Italian regions of Lombardia, Marche, and Puglia. Filledcircles refer to the first 30 days of the epidemic diffusion. The vertical dashed lines markthe times when the Italian government applied lock-down restrictions on 23 February, 01March, 09 March, and 22 March, respectively.

Thus, we selected to use the generalized logistic distribution, also127

considering that its parameters can be linked (in a non-explicit128

way) to the solution of compartmental models as the Susceptible-129

Exposed-Infected-Recovered (SEIR) model and their variants [25]130

or on agent-based models [26]. The generalized logistic distribution for131

fitting the total cumulative number of infections reads [35, 33, 36]132

C(t) =α

1 + β e−γt(1)

being α, β, and γ the parameters of the model. They can be fitted, e.g., using133

Nonlinear least-squares solver, with the Levenberg-Marquardt algorithm and134

the bisquare weight methods to minimize a weighted sum of squares. Here135

we use a MATLAB function to perform the fits. As recently pointed out in136

[27] in the early stages of the epidemics, the smoothness of COVID-19 cumu-137

lative infections data could lead to very uncertain predictions although with138

very good R2. To avoid this, here we focus only on Chinese and Italian data,139

that, to date, represent a mature stage of the epidemics. This implies, as we140

will show, that the significance of the logistic fit can be assigned with greater141

6

confidence [27]. We remark however, that when confinement measures are142

applied, the basic reproduction number R0, which regulates the growth of143

infections, is reduced [37]. We are therefore in presence not of a single lo-144

gistic distribution, but of a mixture of distributions with control parameters145

changing in time as different phases of epidemic diffusion are reached. Con-146

finement measures can reduce R0 from the exponential-like behavior of an147

uncontrolled growing phase, to a smoother logistic growth phase. Our goal148

here is to use the a-priori knowledge of the introduction of confinement mea-149

surements to investigate the perfomrance of statistical prediction of infection150

counts for different epidemic phases. Thus, we perform logistic fits as in151

Eq. (1) in the following time intervals:152

• the first 30 days of epidemic growth, as reported in Figs. 3-4 by black153

lines, thus to consider how restrictions measure globally affect the dif-154

fusion;155

• the first 7 days, roughly corresponding to the time interval during which156

first restriction measures are adopted both in China and Italy, although157

not still completely efficient (red lines in Figs. 3-4);158

• the first 14 days, corresponding to the time interval in which the initial159

confinement measures should lead the first effects (blue lines in Figs. 3-160

4);161

• the time interval between the 8th and the 14th day to investigate how162

the epidemic would be grown if starting from initial restrictions (green163

lines in Figs. 3-4);164

• the time interval between the 15th and the 30th day to investigate the165

efficiency of restriction measures (magenta lines in Figs. 3-4).166

In this way we can investigate both the efficiency of restriction measures167

in containing epidemic growth as well as the stability of prediction models168

based on logistic distribution fitting procedures. Moreover, to assess the169

significance of fits we assume that the last point of the fitting range could be170

affected by a ±30% error. This allows us to provide a simple way to estimate171

confidence intervals for our fits [27]. Finally, the Kolmogorov-Smirnov (K-S)172

test [38, 39, 40] is also used to obtain a test decision for the null hypothesis173

that the observed data are from the same logistic distribution as derived174

from the logistic fits under the different time intervals. This allows to test175

7

the efficiency in delivering reliable forecasts at different stages of the epidemic176

growth. The test is based on evaluating the maximum distance between the177

empirical distribution functions coming from two different samples x1,n and178

x2,m, being n and m the length of samples. By defining the Kolmogorov-179

Smirnov statistic as180

Dn,m = supx|F1,n(x)− F2,m(x)| , (2)

where F1,n(x) and F2,m(x) are the empirical distribution functions of the two181

samples, respectively, the null hypothesis is rejected at the confidence level182

α if183

Dn,m > c(α)

√n+m

n ·m. (3)

When m = n a general relation can be found for Dn(α) as184

Dn(α) >1√n

√− log

(α2

). (4)

The value of c(α) for the most common levels of α are reported in Table 1.

α 0.20 0.15 0.10 0.05 0.01c(α) 1.073 1.138 1.224 1.358 1.628

Table 1: The value of c(α) for the most common levels of α.

185

The closer the observed statistics Dn,obs is to 0 the more likely it is that186

the two samples were drawn from the same distribution with being Dn,obs <187

Dn(α). The use of the K-S test has two main advantages: i) the distribution188

of the K-S test statistic itself does not depend on the underlying cumulative189

distribution function being tested, and ii) it is an exact test [41, 42, 43, 44].190

Moreover, it is specifically designed for testing if data come from a normal,191

a log-normal, a Weibull, an exponential, or a logistic distribution [42, 45].192

Thus, it is particularly suitable for our investigations, being also a non-193

parametric and robust technique since it is not based on strong distributional194

assumptions [42, 45, 46, 44].195

4. Epidemic diffusion through Chinese provinces196

Fig. 3 shows logistic fits for different phases of epidemic across Chinese197

provinces, together with upper and lower confidence bounds, obtained as out-198

8

lined in the previous section. Early stage of epidemic propagation is charac-199

terized by a larger confidence interval (red lines in Fig. 3), thus highlighting200

the difficulty in making early reliable predictions of epidemic growth, with201

an exponential-like behavior. The confidence interval becomes narrower as202

the growth rate reduces, as for the case of the provinces of Bejing and Yun-203

nan being less affected from COVID-19 infections with respect to the Hubei,204

the latter mostly contributing to the overall epidemic growth in China. The205

logistic fit becomes more stable, being characterized by a narrower estimates206

of confidence intervals, when the first two weeks are considered (blue lines207

in Fig. 3), possibly related to the initial efficiency of restriction measures.208

This could be also due to both the limited number of points of the fitting209

range as well as to the particular phase of the epidemic growth. However, by210

comparing the confidence intervals of logistic fits performed using the first211

week (22/01 - 29/01, red lines in Fig. 3) and the second week (30/01 - 05/02,212

green lines in Fig. 3) it is possible to note that the stability increases for this213

second interval for all Chinese provinces, thus suggesting that estimates are214

significantly dependent on the particular epidemic phase considered. Indeed,215

the stability significantly increases when the logistic fit is performed on time216

intervals that do not include the first week of the epidemic growth (green217

and magenta lines in Fig. 3), suggesting that credible predictions could be218

assigned with a large confidence by means of a logistic fit if the beginning219

of the outbreak is not considered. However, the narrowest estimates of sig-220

nificance levels is obtained when the first 30 days are considered, thus also221

including the beginning of the outbreak, possibly suggesting that fits become222

more and more stable if data are collected at a mature stage of the epidemic223

growth. This is clearly visible for all Chinese provinces, apart for the slight224

discrepancy observed for the Bejing province where some returned cases from225

outside China were observed from 20 March. Finally, we assess the statisti-226

cal discrepancy of the logistic fits from the observed data by performing the227

Kolmogorov-Smirnov (K-S) test those results for the 95% confidence level228

are reported in Table 2.229

It can be noted that the statistical results obtained through the K-S test230

suggest that the fits performed by considering the time intervals from 22231

January to 21 February as well as from 05 February to 21 February are sta-232

tistically significant for reproducing the behavior of the observed number233

of infections at the 95% significance level. This seems to support the that234

reliable predictions can be assessed only when a mature stage of the epi-235

demic growth is approached/reached, while low-significant predictions can236

9

Dn,obs

Time interval China Hubei Bejing Yunnan22/01 - 29/01 0.750 0.750 0.750 0.62522/01 - 05/02 0.500 0.475 0.550 0.45030/01 - 05/02 0.575 0.575 0.550 0.52505/02 - 21/02 0.225 0.150 0.125 0.12522/01 - 21/02 0.100 0.100 0.100 0.100

Table 2: Results of the Kolmogorov-Smirnov test for the 95% confidence level for theChinese provinces. The decision to reject the null hypothesis is based on comparingthe observed statistics Dn,obs with the theoretical value Dn,th = 0.2329 obtained for thesignificance level α = 0.05 as in Eq. 4. If Dn,obs < Dn,th then the samples come from thesame logistic distribution and corresponding values are reported in bold.

be released at the early stages of the epidemic diffusion.237

5. Epidemic diffusion through Italian regions238

Fig. 4 shows logistic fits for different phases of epidemic across Italian239

regions, together with the upper and lower confidence lines. As for Chinese240

provinces the early stage of epidemic diffusion is characterized by a larger241

confidence interval (red lines in Fig. 4), again suggesting that reliable predic-242

tions of epidemic growth are particularly difficult in its early stages. Indeed,243

an exponential-like behavior is found for both the Italian territory and Lom-244

bardia, the latter being the first Italian region characterized by COVID-19245

infections. As for China, confidence intervals become narrower as the growth246

rate reduces (see for example Marche or Puglia with respect to Lombardia),247

with the logistic fits also becoming more stable when the initial stages of the248

outbreak are removed (green and magenta lines in Fig. 4). Unlike for Chinese249

regions, Italian regions present a wide range of different epidemic behaviors,250

that we investigate separately in the following.251

5.1. Epidemics growth in Lombardia252

As discussed above the initial phase is characterized by larger uncertain-253

ties and by an exponential-like behavior (red lines in Fig. 4), thus suggesting254

a clear difficulty in making predictions of the growth in its early stage. When255

the first two weeks (e.g., 24/02 - 08/03) are considered (blue lines in Fig. 4) a256

larger uncertainty is found, especially for the upper-bound confidence level.257

This underline the difficulty in making reliable estimates of its evolution.258

10

Similarly, the logistic fits performed between 01 March and 08 March (green259

lines in Fig. 4) suggest that the first two weeks were particularly critical in260

Lombardia, while logistic fits become more stable when removing the be-261

ginning of the outbreak, leading to more reliable estimates of the epidemic262

growth (magenta lines in Fig. 4). Finally, significance levels become narrower263

when the first 30 days are considered (e.g., 24/02 - 23/03), thus also includ-264

ing the beginning of the outbreak, possibly again suggesting that including265

data from the mature stage of the epidemic growth could allow to obtain266

more stable fits. We remark that, no matter the approach followed, logistic267

fits struggle to predict the number of infections of the successive days. This268

failure of statistical real-time forecasts of the epidemics could be related to269

all those factors that can change the instantaneous value of R0, e.g., extended270

violations of the restriction measures, changes in testing protocols or delay271

in data reporting, changes in the virus characteristics. It is worthwhile to272

note that the above features are found for all Northern regions firstly affected273

from COVID-19 diffusion (not shown here).274

5.2. Epidemics growth in Marche275

The epidemic growth throughout Marche, as well as throughout other276

Central regions (not shown), is different from Northern regions. Indeed, the277

first 7 days (e.g., 24/02 - 01/03, red lines in Fig. 4) were not characterized by278

an exponential increase of infections, as the diffusion of the virus was pretty279

slow: logistic fits are therefore meaningless in this context. The exponential280

phase started in the second week, as we can see by fitting the first two week281

of the infection counts (e.g., 24/02 - 08/03, blue lines in Fig. 4) or just the282

second week (e.g., from 01 March to 08 March, green lines in Fig. 4). During283

this week, the number of infections significantly increases (272 confirmed284

cases) enabling better fits of data to logistic distributions. This suggests a285

time delayed propagation between Northern and Central regions. Indeed,286

the logistic fits become more stable, with narrower estimates of confidence287

intervals, when the time interval from 08 March to 23 March (magenta lines in288

Fig. 4) or the first 30 days (e.g., 24/02 - 23/03, black lines in Fig. 4) are taken289

into account, suggesting that credible predictions could be assigned with a290

large confidence when a mature stage of the epidemic growth is approached.291

However, as for Norther regions the logistic fits struggle to predict the number292

of infections of the successive days (i.e., after the first 30 days).293

11

5.3. Epidemics growth in Puglia294

A completely different scenario is found for Puglia and Southern regions295

(not shown). Logistic fits cannot be performed during during the first two296

weeks (e.g., from 24 February to 08 March), as the infection counts was297

not yet exponential. By considering the time interval between 08 and 23298

March (magenta lines in Fig. 4) and the first 30 days (e.g., 24/02 - 23/03,299

black lines in Fig. 4) an increase in the confidence of logistic fits is found,300

although they struggle to predict the number of infections of the successive301

days (i.e., after the first 30 days). This is possibly due to the time delayed302

propagation of epidemic throughout Southern regions for which a mature303

stage is, to date, not yet reached. To support this hypothesis and to assess the304

statistical discrepancy of the logistic fits from the observed data we perform305

the Kolmogorov-Smirnov (K-S) test those results for the 95% confidence level306

are reported in Table 3.

Dn,obs

Time interval Italy Lombardia Marche Puglia24/02 - 01/03 0.825 0.800 0.800 0.80024/02 - 08/03 0.575 0.550 0.650 0.80001/03 - 08/03 0.550 0.425 0.600 0.80008/03 - 23/03 0.325 0.325 0.400 0.40024/02 - 23/03 0.350 0.325 0.400 0.400

Table 3: Results of the Kolmogorov-Smirnov test for the 95% confidence level for the Ital-ian regions. The decision to reject the null hypothesis is based on comparing the observedstatistics Dn,obs with the theoretical value Dn,th = 0.3037 obtained for the significancelevel α = 0.05 as in Eq. 4. If Dn,obs < Dn,th then the samples come from the same logisticdistribution and corresponding values are reported in bold.

307

It is interesting to note that, although lower values of Dn,obs are observed308

when a more mature stage of the epidemic growth is considered in the fitting309

range, as for example for time intervals from 24 February to 23 March as well310

as from 08 to 23 March, the observed values Dn,obs are all above the statistical311

threshold of Dn,th = 0.3037. This suggests that a mature stage is, to the date312

of 23 March, not yet reached, although Northern and Central regions are313

characterized by lower values than the Southern ones, thus possibly related314

to the time delayed propagation of epidemic throughout Southern regions.315

12

6. Estimation of infections for Italy and their peak time316

As discussed in Section 5 all performed logistic fits struggle to predict the317

number of infections of the successive days (i.e., after the first 30 days), thus318

we performed and compare logistic fits in three time intervals: (i) the first 30319

days (e.g., from 24 February to 23 March), (ii) the first 37 days (e.g., from 24320

February to 30 March), and (iii) the overall period from 24 February to 02321

April. The results of the Kolmogorov-Smirnov test for the 95% confidence322

level are reported in Table 4, while the behavior of logistic fits are shown in323

Fig. 5.

Dn,obs

Time interval Italy Lombardia Marche Puglia24/02 - 23/03 0.350 0.325 0.400 0.40024/02 - 30/03 0.150 0.150 0.250 0.27524/02 - 02/04 0.100 0.100 0.175 0.200

Table 4: Results of the Kolmogorov-Smirnov test for the 95% confidence level for theItalian regions. The decision to reject the null hypothesis is based on comparing theobserved statistics Dn,obs with the theoretical value Dn,th = 0.3037 obtained for the sig-nificance level α = 0.05. If Dn,obs < Dn,th then the samples come from the same logisticdistribution.

324

It is interesting to note that all regions and Italy are characterized by325

lower values of Dn,obs, below the theoretical value Dn,th = 0.3037, when326

including the next 7 days (e.g., by considering the period between 24 Febru-327

ary and 30 March) to the logistic fits and when considering the whole time328

range (e.g., 24/02 - 02/04). Lombardia presents lower values of the K-S329

statistics Dn,obs than those for Marche and Puglia, together with a nar-330

rower confidence interval when including the successive days, not observed331

for both Marche and Puglia. Particularly for Puglia the confidence interval332

remains practically unchanged, thus suggesting that logistic fits are not still333

stable, possibly due to the fact that Southern regions have not yet reached334

a mature stage of the epidemic growth. This difference in terms of sta-335

bility of logistic fits as well as on confidence of reliable estimates can be336

clearly seen by looking at the behavior of estimated daily increments. Days337

of peak significantly depends on the fitting range for Puglia, while the esti-338

mation of this quantity is more stable for Lombardia and Marche, as shown339

in Fig. 6. Indeed a wider discrepancy is found between daily increments340

13

and estimates for logistic fits performed during the three intervals, obvi-341

ously affecting both the peak time estimation and its value. By comparing342

our estimates and data collected from the daily report of the Italian Pro-343

tezione Civile (https://github.com/pcm-dpc/COVID-19) we found that the344

discrepancy significantly increases when moving from Northern to Southern345

regions, where it can also reach an error which is comparable with the pre-346

dicted value. This could be the reflection of at least two different factors: i)347

the epidemic growth is in a more mature phase in the Northern and Central348

regions with respect to the Southern ones, where it began with a time delay349

ranging from 3 to 14 days, and ii) the higher ratio between the observed350

cases and the number of tests carried out for Southern regions with respect351

to the rest of Italy (see https://github.com/pcm-dpc/COVID-19). These352

two factors could affect the performance of the logistic fits for the Southern353

regions of Italy, being characterized by wider uncertainties with respect to354

the rest of Italy. Thus, our results suggest that estimates of the ending of355

epidemic growth are affected by the statistical uncertainties, by the delayed356

propagation of infections through the different regions, and by the effective357

respect of the guidelines in terms of confinement measures.358

7. Conclusion359

In this paper we investigated the behavior of predictions of COVID-19360

infections on the particular phase of its growth and propagation in a specific361

country, as well as, on the effectiveness of social distancing and confinement362

measures. By analyzing the epidemic evolution in China and Italy we find363

that predictions are characterized by large uncertainties at the early stages of364

the epidemic growth, significantly reducing when the epidemics peak is past,365

independently on how this is reached. While infection counts for different366

Chinese provinces show a synchronised behavior, counts for Italian regions367

point to different epidemic phases. While the epidemic peak has been likely368

reached in the Northern and Central regions, COVID-19 infections are still369

in a growing phase for Southern regions, with a delay ranging from 3 to 14370

days. By assessing the performance of logistic fits we assess that a wider371

uncertainty is found during the first week of epidemic propagation. Uncer-372

tainty is reduced when data from the very beginning of the breakout are373

removed from the datasets. Moreover, the estimated infection increments374

are extremely sensitive to the epidemic growth stage and to the last points375

considered to perform statistical extrapolations. Higher significance levels376

14

are reached for the more mature stages of the epidemic growth.377

The most interesting pattern in the time-evolution of the dis-378

tribution is the observed change from an exponential-like behavior379

observed at the beginning of the epidemic growth to a sigmoid-like380

one when first restriction measures are introduced, particularly381

evident for the Italian case study. Indeed, by evaluating the ex-382

pected final number of total infections as predicted from logistic fits383

during the different stages we highlight that reliable estimates can-384

not be released until more mature stages of the epidemic growth385

are reached. We show that by only means of the first 7 days,386

corresponding to the time interval during which first restriction387

measures are adopted both in China and Italy, an overestimation388

of the final number of infections of ∼65% for China and ∼2000%389

for Italy is observed. Conversely, by considering the first 14 days,390

corresponding to the time interval in which the initial confinement391

measures should lead the first effects, an underestimation of ∼-48%392

for China and ∼-76% for Italy is obtained. A lower underestimation393

(∼-32% and ∼-69% for China and Italy, respectively) is found when394

considering the time interval between the 8th and the 14th day, e.g.,395

by investigating how the epidemic would be grown if starting from396

initial restrictions only, while a better agreement is found when397

considering the time interval between the 15th and the 30th day,398

corresponding to investigate the efficiency of restriction measures,399

with reduced underestimation of the final number of infections of400

∼-17% for China and ∼-12% for Italy. Finally, by monitoring the401

stability of logistic fits as well as their suitability on predicting402

the number of infections of the successive days (i.e., after the first403

30 days) we highlight how the uncertainty evolution can be used404

to track how the epidemics diffused at a regional level, allowing405

an estimation of the delay in the spread of the virus. Indeed,406

we found that the uncertainty significantly increases when mov-407

ing from Northern to Southern regions, where the error is almost408

comparable with the predicted value, suggesting that, to date, the409

epidemic peak has not been likely reached for Southern regions,410

being delayed with respect to Northern and Central ones.411

Our results aim at providing some guidelines for real-time epidemics fore-412

casts which should be applicable to other viruses and outbreaks. Real-time413

forecasts of the epidemics are, to date, a societal need more than a scientific414

15

field. They are crucial to plan the duration of confinement measures and to415

define the needs for health-care facilities. The aim of this letter was to show416

that those extrapolations crucially depend not only on the quality of data,417

but also on the stage of the epidemics, due to the intrinsically non-linear418

nature of the underlying dynamics. This prevents from performing success-419

ful long-term extrapolations of the infection counts with statistical models.420

As a guideline it is surely helpful to perform logistic fits every421

day and to evaluate the reliability on predicting the next day, and422

then perform a new logistic fit to investigate how the uncertainty423

grown/reduced. Moreover, reliable estimates are surely affected424

by possible source of errors in counting infections, thus we suggest425

to assess the significance of fits to the last data point of the fit-426

ting range by assuming it could be affected by a ±30% error. This427

allows us to provide a simple way to estimate confidence intervals428

[27]. Furthermore, we also suggest not only to exclude the last data429

point and check fits stability but also to consider to exclude initial430

point(s) to evaluate how epidemic would be grown if starting from431

initial restriction measures or how delayed propagation could be432

present at a regional level.433

Our approach, based on a sort of Bayesian framework to update434

the probability for a reduced uncertainty as more evidence or infor-435

mation become available (this especially true for unknown viruses436

and outbreaks), suggests that the statistical modeling of epidemic437

growth should be focused on specific stages of its evolution on time438

as well as on its spread at a more local level (e.g., regional level).439

This can help in controlling local diffusion of epidemics and to re-440

strict the analysis on specific regions depending on its uncertainty441

values. The above guidelines can be also suitable for dynamical mod-442

els such as those based on compartments or agent dynamics which need to be443

initialized with quality data, faithfully representing the infected populations444

including asymptomatic patients [27]. It is therefore crucial to pursue na-445

tional health systems to provide the most transparent and extended datasets446

as possible and obtain high quality datasets to initialize those models. We447

remind that only dynamical models can provide a coherent representation448

and evolution of the epidemics, as they are effectively based on the conser-449

vation of the total number of individuals. Characterizing and modeling450

the uncertainty can allow to preserve the public health and help451

to enforce/relax strict confinement measures.452

16

Acknowledgments453

The authors thank the anonymous reviewer for fruitful and insightful454

comments.455

References456

[1] R. F. Service, Coronavirus epidemic snarls science worldwide, Science457

367 (2020) 836–837. doi:10.1126/science.367.6480.836.458

[2] C. Huang, Y. Wang, X. Li, L. Ren, J. Zhao, Y. Hu, L. Zhang, G. Fan,459

J. Xu, X. Gu, et al., Clinical features of patients infected with 2019460

novel coronavirus in Wuhan, China, The Lancet 395 (2020) 497–506.461

doi:10.1016/S0140-6736(20)30183-5.462

[3] W.-j. Guan, Z.-y. Ni, Y. Hu, W.-h. Liang, C.-q. Ou, J.-x. He, L. Liu,463

H. Shan, C.-l. Lei, D. S. Hui, B. Du, L.-j. Li, G. Zeng, K.-Y. Yuen,464

R.-c. Chen, C.-l. Tang, T. Wang, P.-y. Chen, J. Xiang, S.-y. Li, J.-465

l. Wang, Z.-j. Liang, Y.-x. Peng, L. Wei, Y. Liu, Y.-h. Hu, P. Peng,466

J.-m. Wang, J.-y. Liu, Z. Chen, G. Li, Z.-j. Zheng, S.-q. Qiu, J. Luo,467

C.-j. Ye, S.-y. Zhu, N.-s. Zhong, Clinical characteristics of coronavirus468

disease 2019 in china, New England Journal of Medicine 0 (0) null.469

doi:10.1056/NEJMoa2002032.470

[4] W. H. Organization, Pneumonia of unknown cause471

– china, 2020. URL: https://www.who.int/csr/don/472

05-january-2020-pneumonia-of-unkown-cause-china/en/.473

[5] Q. Li, X. Guan, P. Wu, X. Wang, L. Zhou, Y. Tong, R. Ren, K. S. Leung,474

E. H. Lau, J. Y. Wong, X. Xing, N. Xiang, Y. Wu, C. Li, Q. Chen, D. Li,475

T. Liu, J. Zhao, M. Liu, W. Tu, C. Chen, L. Jin, R. Yang, Q. Wang,476

S. Zhou, R. Wang, H. Liu, Y. Luo, Y. Liu, G. Shao, H. Li, Z. Tao,477

Y. Yang, Z. Deng, B. Liu, Z. Ma, Y. Zhang, G. Shi, T. T. Lam, J. T.478

Wu, G. F. Gao, B. J. Cowling, B. Yang, G. M. Leung, Z. Feng, Early479

transmission dynamics in wuhan, china, of novel coronavirus–infected480

pneumonia, New England Journal of Medicine 382 (2020) 1199–1207.481

doi:10.1056/NEJMoa2001316.482

[6] M. Lipsitch, D. L. Swerdlow, L. Finelli, Defining the epidemiology of483

covid-19 — studies needed, New England Journal of Medicine 382 (2020)484

1194–1196. doi:10.1056/NEJMp2002125.485

17

[7] N. Zhu, D. Zhang, W. Wang, X. Li, B. Yang, J. Song, X. Zhao, B. Huang,486

W. Shi, R. Lu, P. Niu, F. Zhan, X. Ma, D. Wang, W. Xu, G. Wu, G. F.487

Gao, W. Tan, A novel coronavirus from patients with pneumonia in488

china, 2019, New England Journal of Medicine 382 (2020) 727–733.489

doi:10.1056/NEJMoa2001017.490

[8] D. Wang, B. Hu, C. Hu, F. Zhu, X. Liu, J. Zhang, B. Wang,491

H. Xiang, Z. Cheng, Y. Xiong, Y. Zhao, Y. Li, X. Wang,492

Z. Peng, Clinical Characteristics of 138 Hospitalized Pa-493

tients With 2019 Novel Coronavirus–Infected Pneumonia in494

Wuhan, China, JAMA 323 (2020) 1061–1069. URL: https:495

//doi.org/10.1001/jama.2020.1585. doi:10.1001/jama.2020.1585.496

arXiv:https://jamanetwork.com/journals/jama/articlepdf/2761044/jama wang 2020 oi 200019.pdf.497

[9] L. T. Phan, T. V. Nguyen, Q. C. Luong, T. V. Nguyen, H. T.498

Nguyen, H. Q. Le, T. T. Nguyen, T. M. Cao, Q. D. Pham, Im-499

portation and human-to-human transmission of a novel coronavirus500

in vietnam, New England Journal of Medicine 382 (2020) 872–874.501

doi:10.1056/NEJMc2001272, pMID: 31991079.502

[10] W. H. Organization, Coronavirus disease 2019 (covid-19) situ-503

ation report-10, 2020. URL: https://www.who.int/emergencies/504

diseases/novel-coronavirus-2019/situation-reports/.505

[11] W. E. Parmet, M. S. Sinha, Covid-19 — the law and limits of quar-506

antine, New England Journal of Medicine 0 (0) null. doi:10.1056/507

NEJMp2004211.508

[12] R. L. Haffajee, M. M. Mello, Thinking globally, acting locally — the509

u.s. response to covid-19, New England Journal of Medicine 0 (0) null.510

doi:10.1056/NEJMp2006740.511

[13] S. Flaxman, et al., Estimating the number of infections and the512

impact of non-pharmaceutical interventions on covid-19 in 11513

european countries, 2020. URL: https://www.imperial.ac.uk/514

media/imperial-college/medicine/sph/ide/gida-fellowships/515

Imperial-College-COVID19-Europe-estimates-and-NPI-impact-30-03-2020.516

pdf.517

18

[14] C. Rothe, M. Schunk, P. Sothmann, G. Bretzel, G. Froeschl, C. Wall-518

rauch, T. Zimmer, V. Thiel, C. Janke, W. Guggemos, M. Seil-519

maier, C. Drosten, P. Vollmar, K. Zwirglmaier, S. Zange, R. Wolfel,520

M. Hoelscher, Transmission of 2019-ncov infection from an asymp-521

tomatic contact in germany, New England Journal of Medicine 382522

(2020) 970–971. doi:10.1056/NEJMc2001468.523

[15] S. A. Lauer, K. H. Grantz, Q. Bi, F. K. Jones, Q. Zheng, H. R. Meredith,524

A. S. Azman, N. G. Reich, J. Lessler, The Incubation Period of Coro-525

navirus Disease 2019 (COVID-19) From Publicly Reported Confirmed526

Cases: Estimation and Application, Annals of Internal Medicine (2020).527

URL: https://doi.org/10.7326/M20-0504. doi:10.7326/M20-0504.528

arXiv:https://annals.org/acp/content public/journal/aim/0/aime202005050-m200504.pdf.529

[16] P. M. of Italy, Disposizioni attuative del decreto-legge 23 febbraio530

2020, n. 6, recante misure urgenti in materia di contenimento e ges-531

tione dell’emergenza epidemiologica da covid-19, 2020. URL: https:532

//www.gazzettaufficiale.it/eli/id/2020/02/23/20A01228/sg.533

[17] P. M. of Italy, Ulteriori disposizioni attuative del decreto-legge 23 feb-534

braio 2020, n. 6, recante misure urgenti in materia di contenimento e ges-535

tione dell’emergenza epidemiologica da covid-19, applicabili sull’intero536

territorio nazionale, 2020. URL: https://www.gazzettaufficiale.537

it/eli/id/2020/03/09/20A01558/sg.538





it/eli/id/2020/03/11/20A01605/sg.543





it/eli/id/2020/03/22/20A01807/sg.548

[20] L. Rosenbaum, Facing covid-19 in italy — ethics, logistics, and thera-549

peutics on the epidemic’s front line, New England Journal of Medicine550

0 (0) null. doi:10.1056/NEJMp2005492.551

19

[21] M. Chinazzi, J. T. Davis, M. Ajelli, C. Gioannini, M. Litvi-552

nova, S. Merler, A. Pastore y Piontti, K. Mu, L. Rossi, K. Sun,553

C. Viboud, X. Xiong, H. Yu, M. E. Halloran, I. M. Longini,554

A. Vespignani, The effect of travel restrictions on the spread555

of the 2019 novel coronavirus (covid-19) outbreak, Science556

(2020). URL: https://science.sciencemag.org/content/early/557

2020/03/05/science.aba9757. doi:10.1126/science.aba9757.558

arXiv:https://science.sciencemag.org/content/early/2020/03/05/science.aba9757.full.pdf.559

[22] A. S. Fauci, H. C. Lane, R. R. Redfield, Covid-19 — navigating the560

uncharted, New England Journal of Medicine 382 (2020) 1268–1269.561

doi:10.1056/NEJMe2002387.562

[23] W. H. Organization, Coronavirus disease 2019 (covid-19)563

situation report-51, 2020. URL: https://www.who.int/564

docs/default-source/coronaviruse/situation-reports/565

20200311-sitrep-51-covid-19.pdf?sfvrsn=1ba62e57\_10.566

[24] A. Spinello, A. Saltalamacchia, A. Magistrato, Is the Rigid-567

ity of SARS-CoV-2 Spike Receptor-Binding Motif the Hall-568

mark for Its Enhanced Infectivity and Pathogenesis? (2020).569

URL: https://chemrxiv.org/articles/Is_the_Rigidity_of_570

SARS-CoV-2_Spike_Receptor-Binding_Motif_the_Hallmark_571

for_Its_Enhanced_Infectivity_and_Pathogenesis_/12091260.572

doi:10.26434/chemrxiv.12091260.v1.573

[25] F. Brauer, Compartmental models in epidemiology, Lecture Notes in574

Mathematics, Springer Verlag, 2008, pp. 19–79.575

[26] S. L. Chang, N. Harding, C. Zachreson, O. M. Cliff, M. Prokopenko,576

Modelling transmission and control of the covid-19 pandemic in aus-577

tralia, arXiv preprint arXiv:2003.10218 (2020).578

[27] D. Faranda, I. P. Castillo, O. Hulme, A. Jezequel, J. S. W. Lamb,579

Y. Sato, E. L. Thompson, Asymptotic estimates of sars-cov-2 infection580

counts and their sensitivity to stochastic perturbation, Chaos under581

review (2020).582

[28] T. C. Wilcosky, L. E. Chambless, A comparison of direct adjustment583

and regression adjustment of epidemiologic measures, Journal of chronic584

diseases 38 (1985) 849–856.585

20

[29] M. Batista, Estimation of the final size of the covid-19 epidemic,586

Preprint.] medRxiv (2020).587

[30] C.-H. Li, C.-C. Tsai, S.-Y. Yang, Analysis of the permanence of an588

SIR epidemic model with logistic process and distributed time delay,589

Communications in Nonlinear Science and Numerical Simulations 17590

(2012) 3696–3707. doi:10.1016/j.cnsns.2012.01.018.591

[31] J. Kumar, K. P. S. S. Hembram, Epidemiological study of novel592

coronavirus (COVID-19), arXiv e-prints (2020) arXiv:2003.11376.593

arXiv:2003.11376.594

[32] J. Ma, Estimating epidemic exponential growth rate and basic re-595

production number, Infectious Disease Modelling 5 (2020) 129 –596

141. URL: http://www.sciencedirect.com/science/article/pii/597

S2468042719300491. doi:https://doi.org/10.1016/j.idm.2019.12.598

009.599

[33] G. Chowell, Fitting dynamic models to epidemic outbreaks with600

quantified uncertainty: A primer for parameter uncertainty, identi-601

fiability, and forecasts, Infectious Disease Modelling 2 (2017) 379602

– 398. URL: http://www.sciencedirect.com/science/article/pii/603

S2468042717300234. doi:https://doi.org/10.1016/j.idm.2017.08.604

001.605

[34] I. J. Wellock, G. C. Emmans, I. Kyriazakis, Describing and predicting606

potential growth in the pig, Animal Science 78 (2004) 379–388. doi:10.607

1017/S1357729800058781.608

[35] B. Gompertz, On the Nature of the Function Expressive of the Law609

of Human Mortality, and on a New Mode of Determining the Value of610

Life Contingencies, Philosophical Transactions of the Royal Society of611

London Series I 115 (1825) 513–583.612

[36] R. Burger, G. Chowell, L. Y. Lara-Dııaz, Comparative analysis of phe-613

nomenological growth models applied to epidemic outbreaks, Mathe-614

matical biosciences and engineering : MBE 16 (2019) 4250—4273. URL:615

https://doi.org/10.3934/mbe.2019212. doi:10.3934/mbe.2019212.616

[37] G. Viceconte, N. Petrosillo, Covid-19 r0: Magic number or co-617

nundrum?, Infectious Disease Reports 12 (2020). URL: https://618

21

www.pagepress.org/journals/index.php/idr/article/view/8516.619

doi:10.4081/idr.2020.8516.620

[38] A. N. Kolmogorov, Sulla determinazione empirica di una legge di dis-621

tribuzione, Giornale dell’Istituto Italiano degli Attuari 4 (1933) 83–91.622

[39] N. Smirnov, Table for estimating the goodness of fit of empirical distri-623

butions, Ann. Math. Statist. 19 (1948) 279–281. URL: https://doi.624

org/10.1214/aoms/1177730256. doi:10.1214/aoms/1177730256.625

[40] M. A. Stephens, Edf statistics for goodness of fit and some compar-626

isons, Journal of the American Statistical Association 69 (1974) 730–627

737. doi:10.1080/01621459.1974.10480196.628

[41] T. W. Anderson, D. A. Darling, A test of goodness of fit, Journal of629

the American Statistical Association 49 (1954) 765–769. doi:10.1080/630

01621459.1954.10501232.631

[42] G. Marsaglia, W. W. Tsang, J. Wang, Evaluating kolmogorov’s distribu-632

tion, Journal of Statistical Software, Articles 8 (2003) 1–4. URL: https:633

//www.jstatsoft.org/v008/i18. doi:10.18637/jss.v008.i18.634

[43] Kolmogorov–Smirnov Test, Springer New York, New York, NY, 2008,635

pp. 283–287. URL: https://doi.org/10.1007/978-0-387-32833-1_636

214. doi:10.1007/978-0-387-32833-1_214.637

[44] M. Marozzi, Nonparametric simultaneous tests for location and scale638

testing: A comparison of several methods, Communications in Statis-639

tics - Simulation and Computation 42 (2013) 1298–1317. doi:10.1080/640

03610918.2012.665546.641

[45] F. J. Massey, Distribution table for the deviation between two sample642

cumulatives, Ann. Math. Statist. 23 (1952) 435–441. URL: https://643

doi.org/10.1214/aoms/1177729388. doi:10.1214/aoms/1177729388.644

[46] W. Ghidey, E. Lesaffre, G. Verbeke, A comparison of methods for esti-645

mating the random effects distribution of a linear mixed model, Statis-646

tical Methods in Medical Research 19 (2010) 575–600.647

22

0.5

1

1.5

2

Co

nfi

rmed

- C

hin

a

105

0.5

1

1.5

2

Co

nfi

rmed

- H

ub

ei

105

200

400

600

800

1000

Co

nfi

rmed

- B

ejin

g

22/01 - 21/02

22/01 - 29/01

22/01 - 05/02

30/01 - 05/02

05/02 - 21/02

30 Jan 19 Feb 10 Mar 30 Mar

50

100

150

200

250

300

350

400

Co

nfi

rmed

- Y

un

nan

Figure 3: Logistic fits during the different time intervals of epidemic across Chineseprovinces, together with the confidence lines. From top to bottom: China and threeprovinces (Bejing, Hubei, Yunnan). The vertical dashed lines mark the times when Chinesegovernment applied lock-down restrictions on 23 January and 28 January, respectively.

23

2

4

6

8

10

12

14

Con

firm

ed -

Ita

ly

104

24/02 - 23/03

24/02 - 01/03

24/02 - 08/03

01/03 - 08/03

08/03 - 23/03

1

2

3

4

5

Con

firm

ed -

Lom

bard

ia

104

1000

2000

3000

4000

5000

Con

firm

ed -

Mar

che

20 Feb 29 Feb 10 Mar 20 Mar 30 Mar 09 Apr

500

1000

1500

2000

2500

3000

Con

firm

ed -

Pug

lia

Figure 4: Logistic fits during the different time intervals of epidemic across Italian re-gions, together with the confidence lines. From top to bottom: Italy and three regions(Lombardia, Marche, Puglia). The vertical dashed lines mark the times when Italiangovernment applied lock-down restrictions on 23 February, 01 March, 09 March, and 22March, respectively.

24

2

4

6

8

10

12

14

Co

nfi

rmed

- I

taly

104

24/02 - 23/03

24/02 - 30/03

24/02 - 02/04

1

2

3

4

5

Co

nfi

rmed

- L

om

bar

dia

104

1000

2000

3000

4000

5000

Co

nfi

rmed

- M

arch

e


500

1000

1500

2000

2500

3000

Co

nfi

rmed

- P

ug

lia

Figure 5: Logistic fits during the different time intervals of epidemic across Italian re-gions, together with the confidence lines. From top to bottom: Italy and three regions(Lombardia, Marche, Puglia). The vertical dashed lines mark the times when Italiangovernment applied lock-down restrictions on 23 February, 01 March, 09 March, and 22March, respectively.

25

1000

2000

3000

4000

5000

6000

7000

8000In

crem

ents

- I

taly

24/02 - 23/03

24/02 - 30/03

24/02 - 02/04

500

1000

1500

2000

2500

3000

3500

4000

Incr

emen

ts -

Lo

mb

ard

ia

50

100

150

200

250

300

Incr

emen

ts -

Mar

che


50

100

150

200

Incr

emen

ts -

Pu

gli

a

Figure 6: Estimation of daily infections and their peak time during three different timeintervals of epidemic across Italian regions, together with the confidence lines. From top tobottom: Italy and three regions (Lombardia, Marche, Puglia). The vertical dashed linesmark the times when Italian government applied lock-down restrictions on 23 February,01 March, 09 March, and 22 March, respectively.

26

on the uncertainty of real-time predictions of epidemic

Documents